End-to-End Simulation Loop

• The end-to-end simulation loop is a computational workflow that integrates differentiable simulation with gradient-based optimization via coupled inner and outer loops.
• It enables analytic gradient computation with respect to simulation parameters, facilitating property-driven inverse design and multi-objective optimization.
• Applications include force field fitting, molecular dynamics, and ML potential tuning, offering a robust framework for automated materials design.

An end-to-end simulation loop is a computational workflow in which the complete mapping from low-level model parameters or system inputs through dynamic or static simulation to final observables and loss functions is parameterized and differentiated or optimized as a whole. This construct allows for the analytical computation of gradients with respect to simulation parameters, direct loss minimization on high-level target properties, and seamless integration of physical, machine-learned, or hybrid model components. In atomistic simulation, the end-to-end loop is realized by fully differentiable molecular dynamics or statics, automatic differentiation of property computations, and outer-loop gradient-based optimization of force field parameters against arbitrary property losses, yielding a robust platform for inverse design, force field fitting, and multi-objective optimization (Gangan et al., 2024).

1. Formal Structure of the End-to-End Simulation Loop

An end-to-end simulation loop consists of two principal stages: a differentiable inner simulation loop and an outer optimization loop. The inner loop generates time-evolved or static system trajectories under the current parameterization, computes the desired observables, and propagates gradients with respect to all underlying parameters. The outer loop accumulates property-wise losses, aggregates them (possibly with per-property weights), and updates parameter vectors using gradients obtained from the entire simulation workflow.

• Inner simulation loop: Given parameters $\theta \in \mathbb{R}^k$ (force field coefficients or model weights), states $r_t, v_t$ (atomic positions and velocities), and functional forms $U(\theta; r_t)$ (potential energy), the loop propagates the system using a time integrator such as velocity-Verlet:

$$r_{t+1} = r_t + \Delta t\, v_t + \frac{\Delta t^2}{2} m^{-1} F_t, \quad v_{t+1} = v_t + \frac{\Delta t}{2} m^{-1} (F_t + F_{t+1}),$$

where $F_t = -\nabla_{r_t} U(\theta; r_t)$. All quantities are implemented to permit automatic differentiation w.r.t. $\theta$ (Gangan et al., 2024).

• Property evaluation: After $T$ steps, trajectory-dependent observables (e.g., elastic constants, vibrational spectra, radial distribution functions) are computed; their derivatives $\partial$(observable)$/\partial\theta$ are available by backpropagation through all time steps.
• Outer optimization loop: For $P$ target properties, losses $L_p(\theta)$ quantify the distance between simulated $O_p(\theta)$ and reference $O_p^{\rm target}$, with the total loss

$$L_{\rm total}(\theta) = \sum_{p=1}^P w_p\, L_p(\theta).$$

The gradient $\nabla_\theta L_{\rm total}$ is computed by automatic differentiation through the entire simulation. Parameters are updated by gradient-based optimizers (e.g., SGD, Adam, CG, BFGS).

2. Mathematical and Algorithmic Formulation

The inner and outer loops are amenable to formal differentiation by virtue of their analytic construction:

• Analytic gradients via autodiff: Key to the end-to-end loop is analytical computation of derivatives such as

$$\frac{\partial r_{t+1}}{\partial \theta} = \frac{\Delta t^2}{2} m^{-1} \frac{\partial F_t}{\partial \theta} + \cdots.$$

All recursive dependencies (e.g., $F_{t+1}$ depends on $r_{t+1}$, which in turn depends on $\theta$) allow for exact gradient computation by automatic differentiation (Gangan et al., 2024).

• Property losses: Examples include
  • Elastic constant loss:

  $$L_{\rm elastic}(\theta) = \sum_{i,j=1}^6 \bigl[C_{ij}^{\rm sim}(\theta) - C_{ij}^{\rm target}\bigr]^2$$ - Force constant (Hessian) loss:

  $$L_{\rm FC}(\theta) = \sum_{i,j=1}^{3N} [H_{ij}^{\rm sim}(\theta) - H_{ij}^{\rm target}]^2$$

  with $$H_{ij} = \frac{\partial^2 U}{\partial r_i \partial r_j}$$. - Radial distribution function:

  $$L_{\rm RDF}(\theta) = \sum_b [g^{\rm sim}(r_b; \theta) - g^{\rm target}(r_b)]^2$$ - Multi-objective:

  $$L_{\rm total}(\theta) = \sum_{p=1}^P w_p L_p(\theta)$$

• Parameter update: Any gradient-based rule,

$$\theta \leftarrow \theta - \alpha\, \nabla_\theta L_{\rm total}(\theta)$$

• Pseudocode (for one full iteration):

initialize θ ← θ₀ repeat until convergence: for each property p=1…P: compute property O_p(θ) compute loss L_p(θ) aggregate L_total = Σ_p w_p L_p(θ) compute ∇θ L_total via autodiff update θ ← optimizer.update(θ, ∇θ L_total) return θ*

3. Software Implementation and System Integration

The fully end-to-end differentiable simulation loop is realized using scientific differentiable programming platforms:

• JAX/JAX-MD: Implementation utilizes the JAX auto-differentiation (jax.grad, jax.jacfwd, jax.jacrev) and functional programming constructs (jax.jit, jax.vmap) for efficient, compiled simulation and vectorized parameter/trajectory computation (Gangan et al., 2024).
• Functional integrator design: Each simulation step is written as a pure function over JAX arrays, ensuring all intermediate states persist as differentiable tensors.
• Memory management: Backpropagation through $T$ steps requires storage (or checkpointing) of entire trajectory $(r_t, v_t, \dots)_{t=0}^T$. Techniques such as gradient-checkpointing allow memory-vs-computation tradeoffs.
• Property-specific modules: Elastic constants and Hessian matrices are generated using JAX’s second-order autodiff directly on strain or position energy surfaces.

4. Case Studies and Applications

The end-to-end simulation loop framework demonstrates efficacy in both classical and machine-learned force field optimization, as well as in multi-objective and property-driven fitting:

Case	Loss Terms	Parameter Update Outcome
SW/EDIP statics	$L_{\rm elastic}$	$\sim$4–6 iterations, $<5\%$ error on DFT $C_{ij}$
Vibrational fit	$L_{\rm FC}$	Phonons and vibrational DOS near-DFT accuracy
ML potential	$L_{\rm RDF}$	Corrects $g(r)$ at high $T$, generalizes across $T$
Multi-objective	$L_{\rm elastic}$+$L_{\rm FC}$	Pareto compromise, balanced property match

• Silicon system optimization: Using SW and EDIP potentials, starting from canonical parameter values, the end-to-end loop converged in a handful of outer-loop steps to parameters matching ab initio elastic and vibrational properties. Both statics (elastic constants) and dynamics (RDF, vibrational DOS) can be targeted in a unified framework (Gangan et al., 2024).
• Machine-learned models: The framework can fine-tune deep message-passing neural network potentials directly on property observables (e.g., RDF histogram), with analytical gradients through the MD trajectory, achieving better generalization to conditions outside training data than force/energy-only fine-tuning.

5. Generality and Extensibility

The methodology of the end-to-end simulation loop is generically applicable to any simulation where parameters enter analytically into the system evolution and property map:

• Force field families: SW, EDIP, or GNN/ML-based force fields, supporting continuous or high-dimensional parameterizations.
• Physical properties: Any observable that can be written as a function of simulation trajectory or statics, including those requiring time-averaging, spectral analysis, or structural reduction, can serve as a direct loss term.
• Multi-objective:

$L_{\rm total}(\theta) = \sum_p w_p L_p(\theta)$

allows simultaneous targeting of, e.g., $C_{ij}$ and $H_{ij}$, guiding the system to Pareto-efficient solutions.

• Scalability: For moderate parameter dimensionality, losses are typically strongly convex in parameter space, allowing rapid (few-step) convergence. The framework extends naturally to larger models or mixed empirical/ML potentials.

6. Significance and Implications for Inverse Design

The end-to-end simulation loop paradigm enables direct, property-driven inverse design in the physical and materials sciences:

• By coupling analytic parameter gradients to global, physically relevant objectives (rather than force/energy proxies), one can steer models toward accurate, generalizable reproductions of complex material response—including those difficult to fit by traditional energy/force matching (Gangan et al., 2024).
• Differentiable pipelines accelerate force field fitting cycles, facilitate composable multi-property loss definitions, and open up new avenues for automated materials design and simulation-experiment matching, applicable also to higher-level ML potentials.

---

The end-to-end simulation loop represents a general, platform-independent abstraction for automatic, property-driven optimization of simulation parameterizations, as rigorously formulated and implemented in differentiable atomistic simulation for force field fitting and beyond (Gangan et al., 2024).